Comparison Between Direct Mercators and Lamberts Projections
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COMPARISON BETWEEN DIRECT MERCATORS AND LAMBERTS PROJECTIONS
MERCATORS PROJECTION
In the 16th century, a Flemish navigator called Gerhard Kremer, who used the Latin alias Mercator, recognised the limitations of the simple cylindrical projection. The projected graticule met one o f the requirements for an orthomorphic/conformal chart – the meridians and parallels crossed at right angles. However, the shapes were clearly not correct and therefore angles on the chart were not correct.
They are stretched in a N-S direction. Mercator determined that this distortion of angles was caused by failure to meet the second requirement of orthomorphism/conformality – at any point on a chart, scale should be the same in all directions, or should change at the same rate in all directions. On the simple cylindrical projection, the N-S scale was changing at a different rate from the E-W scale. Mercator determined that the E-W scale was changing at a rate proportional to the secant of the latitude (secant = IIcosine). However, the N-S scale was changing at a rate proportional to the square ofthe secant (sec2) of the latitude, resulting in the N-S stretching of shapes.
Mercator solved the problem by adjusting the positions of the parallels of latitude. The parallels had been projected as parallel lines with separation between the parallels increasing (at a rate of sec2 latitude) away from the Equator. Mercator adjusted the parallels of latitude so that their separation increased only at a rate proportional to the secant of Latitude, matching the E-W scale change.
Briefly, Mercator mathematically adjusted the positions of the parallels of latitude to make the chart orthomorphic/conformal.
Because the chart has been mathematically produced, it is a non-perspective chart. This Mercator projection is often called a normal or direct Mercator. The projection surface touches the Reduced Earth at the Equator. The geographic poles cannot be projected( they are on the axis of the cylinder).
(1)
MERCATOR CHARTS – BASIC PROPERTIES
a) Orthomorphism
The chart is orthomorphic/conformal by mathematical construction (Mercators adjustment of the parallels of latitude). The projection is non-perspective.
b) Graticule
The graticule is rectangular. Meridians are equally spaced parallel lines. Parallels of latitude are unequally spaced parallel lines.
c) Rhumb Lines
Because the meridians are parallel lines, a straight line track drawn on the chart will cut all meridians at the same angle. A straight line track on a Mercator chart is a rhumb line.
d) Great Circles
“The Rhumb line between two points will always be nearer to the Equator than the corresponding Great Circle. Conversely, the Great Circle between two points will always lie nearer the Pole than the Rhumb line”.
e) Chart Convergence
Earth Convergency (EC) was defined as the angle of inclination between two selected meridians on the Earth measured at a given latitude. On navigation charts, the equivalent is Chart Convergence (CC). Chart convergence is the angle of inclination between two selected meridians on the chart, measured at a given latitude.
On a Mercator chart, Chart Convergence (CC) is zero all over the chart – (the meridians are parallel).
At the Equator on the Earth, Earth Convergency is zero. Elsewhere on the Earth: Earth Convergency: ch long x sin mean lat
Earth Convergency is always greater than zero except at the Equator. Thus we can say that:
(a) On a Mercator chart at the Equator, Chart Convergence is equalto Earth Convergency is equal to zero.
(b) Chart Convergence on a Mercator Chart is “correct” at the Equator.
(c) At all latitudes apart from the Equator, Chart Convergence on a Mercator chart is less than Earth Convergency.
f) Scale
Scale on a Mercator chart is correct at the Equator. Scale expands away from the Equator. The scale expands at a rate proportional to the secant of latitude.
(2)
LAMBERTS CONICAL ORTHOMORPHIC PROJECTION.
Lamberts Conical Orthomorphic Projection is based on the simple conical projection and is produced mathematically from it.
Firstly, the scale is reduced all over the chart. Since scale on the simple conic is correct only on the standard parallel and scale on either side o f it is greater (expanded), this reduction will give two parallels on which scale is correct; one on either side of the original standard parallel.
Secondly, further mathematical modification